Decision Theory 1 Outline Rational preferences Utilities Money - - PowerPoint PPT Presentation

Decision Theory

1

Outline

♦ Rational preferences ♦ Utilities ♦ Money ♦ Multiattribute utilities ♦ Decision networks ♦ Value of information

2

Preferences

An agent chooses among prizes (A, B, etc.) and lotteries, i.e., situations with uncertain prizes Lottery L = [p, A; (1 − p), B]

L p 1−p A B

Notation: A ≻ B A preferred to B A ∼ B indifference between A and B A ≻ ∼ B B not preferred to A

3

Rational preferences

Idea: preferences of a rational agent must obey constraints. Rational preferences ⇒ behavior describable as maximization of expected utility Constraints: Orderability (A ≻ B) ∨ (B ≻ A) ∨ (A ∼ B) Transitivity (A ≻ B) ∧ (B ≻ C) ⇒ (A ≻ C) Continuity A ≻ B ≻ C ⇒ ∃ p [p, A; 1 − p, C] ∼ B Substitutability A ∼ B ⇒ [p, A; 1 − p, C] ∼ [p, B; 1 − p, C] Monotonicity A ≻ B ⇒ (p ≥ q ⇔ [p, A; 1 − p, B] ≻ ∼ [q, A; 1 − q, B])

4

Rational preferences contd.

Violating the constraints leads to self-evident irrationality For example: an agent with intransitive preferences can be induced to give away all its money If B ≻ C, then an agent who has C would pay (say) 1 cent to get B If A ≻ B, then an agent who has B would pay (say) 1 cent to get A If C ≻ A, then an agent who has A would pay (say) 1 cent to get C

A B C

1c 1c 1c

5

Maximizing expected utility

Theorem (Ramsey, 1931; von Neumann and Morgenstern, 1944): Given preferences satisfying the constraints there exists a real-valued function U such that U(A) ≥ U(B) ⇔ A ≻ ∼ B U([p1, S1; . . . ; pn, Sn]) = Σi piU(Si) MEU principle: Choose the action that maximizes expected utility Note: an agent can be entirely rational (consistent with MEU) without ever representing or manipulating utilities and probabilities E.g., a lookup table for perfect tictactoe

6

Utilities

Utilities map states to real numbers. Which numbers? Standard approach to assessment of human utilities: compare a given state A to a standard lottery Lp that has “best possible prize” u⊤ with probability p “worst possible catastrophe” u⊥ with probability (1 − p) adjust lottery probability p until A ∼ Lp

L 0.999999 0.000001 continue as before instant death

pay $30 ~

7

Utility scales

Normalized utilities: u⊤ = 1.0, u⊥ = 0.0 Micromorts: one-millionth chance of death useful for Russian roulette, paying to reduce product risks, etc. QALYs: quality-adjusted life years useful for medical decisions involving substantial risk Note: behavior is invariant w.r.t. +ve linear transformation U ′(x) = k1U(x) + k2 where k1 > 0 With deterministic prizes only (no lottery choices), only

• rdinal utility can be determined, i.e., total order on prizes

8

Money

Money does not behave as a utility function Given a lottery L with expected monetary value EMV (L), usually U(L) < U(EMV (L)), i.e., people are risk-averse Utility curve: for what probability p am I indifferent between a prize x and a lottery [p, $M; (1 − p), $0] for large M? Typical empirical data, extrapolated with risk-prone behavior:

+U +$

−150,000 800,000

• o o
• o o
• 9

Student group utility

For each x, adjust p until half the class votes for lottery (M=10,000)

p $x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 500 3000 4000 5000 6000 7000 8000 9000 10000 1000 2000

10

Decision networks

Add action nodes and utility nodes to belief networks to enable rational decision making

U

Airport Site

Deaths Noise Cost Litigation Construction Air Traffic

Algorithm: For each value of action node compute expected value of utility node given action, evidence Return MEU action

11

Multiattribute utility

How can we handle utility functions of many variables X1 . . . Xn? E.g., what is U(Deaths, Noise, Cost)? How can complex utility functions be assessed from preference behaviour? Idea 1: identify conditions under which decisions can be made without com- plete identification of U(x1, . . . , xn) Idea 2: identify various types of independence in preferences and derive consequent canonical forms for U(x1, . . . , xn)

12

Strict dominance

Typically define attributes such that U is monotonic in each Strict dominance: choice B strictly dominates choice A iff ∀ i Xi(B) ≥ Xi(A) (and hence U(B) ≥ U(A))

1

X

2

X A B C D

1

X

2

X A B C

This region dominates A

Deterministic attributes Uncertain attributes

Strict dominance seldom holds in practice

13

Stochastic dominance

0.2 0.4 0.6 0.8 1 1.2

• 6
• 5.5
• 5
• 4.5
• 4
• 3.5
• 3
• 2.5
• 2

Probability Negative cost S1 S2 0.2 0.4 0.6 0.8 1

• 6
• 5.5
• 5
• 4.5
• 4
• 3.5
• 3
• 2.5
• 2

Probability Negative cost S1 S2

Distribution p1 stochastically dominates distribution p2 iff ∀ t

t

−∞ p1(x)dx ≤

t

−∞ p2(t)dt

If U is monotonic in x, then A1 with outcome distribution p1 stochastically dominates A2 with outcome distribution p2:

∞

−∞ p1(x)U(x)dx ≥

∞

−∞ p2(x)U(x)dx

Multiattribute case: stochastic dominance on all attributes ⇒

• ptimal

14

Stochastic dominance contd.

Stochastic dominance can often be determined without exact distributions using qualitative reasoning E.g., construction cost increases with distance from city S1 is closer to the city than S2 ⇒ S1 stochastically dominates S2 on cost E.g., injury increases with collision speed Can annotate belief networks with stochastic dominance information: X

+

− → Y (X positively influences Y ) means that For every value z of Y ’s other parents Z ∀ x1, x2 x1 ≥ x2 ⇒ P(Y |x1, z) stochastically dominates P(Y |x2, z)

15

Label the arcs + or –

SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost

16

Label the arcs + or –

SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost

+

17

Label the arcs + or –

SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost

+ +

18

Label the arcs + or –

SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost

+ + −

19

Label the arcs + or –

SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost

+ + − −

20

Label the arcs + or –

SocioEcon Age GoodStudent ExtraCar Mileage VehicleYear RiskAversion SeniorTrain DrivingSkill MakeModel DrivingHist DrivQuality Antilock Airbag CarValue HomeBase AntiTheft Theft OwnDamage PropertyCost LiabilityCost MedicalCost Cushioning Ruggedness Accident OtherCost OwnCost

+ + − −

21

Preference structure: Deterministic

X1 and X2 preferentially independent of X3 iff preference between x1, x2, x3 and x′

1, x′ 2, x3

does not depend on x3 E.g., Noise, Cost, Safety: 20,000 suffer, $4.6 billion, 0.06 deaths/mpm vs. 70,000 suffer, $4.2 billion, 0.06 deaths/mpm Theorem (Leontief, 1947): if every pair of attributes is P.I. of its com- plement, then every subset of attributes is P.I of its complement: mutual P.I.. Theorem (Debreu, 1960): mutual P.I. ⇒ ∃ additive value function: V (S) = ΣiVi(Xi(S)) Hence assess n single-attribute functions; often a good approximation

22

Preference structure: Stochastic

Need to consider preferences over lotteries: X is utility-independent of Y iff preferences over lotteries in X do not depend on y Mutual U.I.: each subset is U.I of its complement ⇒ ∃ multiplicative utility function: U = k1U1 + k2U2 + k3U3 + k1k2U1U2 + k2k3U2U3 + k3k1U3U1 + k1k2k3U1U2U3 Routine procedures and software packages for generating preference tests to identify various canonical families of utility functions

23

Value of information

Idea: compute value of acquiring each possible piece of evidence Can be done directly from decision network Example: buying oil drilling rights Two blocks A and B, exactly one has oil, worth k Prior probabilities 0.5 each, mutually exclusive Current price of each block is k/2 “Consultant” offers accurate survey of A. Fair price? Solution: compute expected value of information = expected value of best action given the information minus expected value of best action without information Survey may say “oil in A” or “no oil in A”, prob. 0.5 each (given!) = [0.5 × value of “buy A” given “oil in A” + 0.5 × value of “buy B” given “no oil in A”] – 0 = (0.5 × k/2) + (0.5 × k/2) − 0 = k/2

24

General formula

Current evidence E, current best action α Possible action outcomes Si, potential new evidence Ej EU(α|E) = max

a Σi U(Si) P(Si|E, a)

Suppose we knew Ej = ejk, then we would choose αejk s.t. EU(αejk|E, Ej = ejk) = max

a Σi U(Si) P(Si|E, a, Ej = ejk)

Ej is a random variable whose value is currently unknown ⇒ must compute expected gain over all possible values: V PIE(Ej) =

Σk P(Ej = ejk|E)EU(αejk|E, Ej = ejk)

• − EU(α|E)

(VPI = value of perfect information)

25

Properties of VPI

Nonnegative—in expectation, not post hoc ∀ j, E V PIE(Ej) ≥ 0 Nonadditive—consider, e.g., obtaining Ej twice V PIE(Ej, Ek) = V PIE(Ej) + V PIE(Ek) Order-independent V PIE(Ej, Ek) = V PIE(Ej) + V PIE,Ej(Ek) = V PIE(Ek) + V PIE,Ek(Ej) Note: when more than one piece of evidence can be gathered, maximizing VPI for each to select one is not always optimal ⇒ evidence-gathering becomes a sequential decision problem

26

Qualitative behaviors

a) Choice is obvious, information worth little b) Choice is nonobvious, information worth a lot c) Choice is nonobvious, information worth little

P ( U | E )

j

P ( U | E )

j

P ( U | E )

j

(a) (b) (c)

U U U U

1

U

2

U

2

U

2

U

1

U

1

27